Schan2, T. & Venneer. P. A. (1996). Giotechtliqul! 46, No. I, 145-151

TECHNICAL NOTE

Angles of friction and dilatancy of sand

T. SCHANZ' and P. A. VERMEER'

KEYWORDS: laboratory tests; plasticit)'; nnds; shearstrength.

INTRODUCTIONThe strength of sand is usually characterized by thepeak friction angle ¢p and the critical state frictionangle ¢Cy. It is generally realized that the peakfriction angel depends not only on density but alsoon the stress path, including differences betweenplane strain and triaxial testing conditions. Indeed,plane strain and triaxial strain angles can differ bymore than 5° for a dense sand. For a loose sand atthe critical density it is often suggesled that similardifferences occur (e.g. Stroud, 1971; Lade, 1984).However, some authors have presented data thatsuggest a unique critical state angle (e.g. Rowe,1962, 1971; Bolton, 1986).

This technical note presents data on a uniquecritical state angle. The implication is that thefailure criterion of a very loose sand is accuratelydescribed by the Mohr--coulomb condition, whichgives the known six-sided pyramid in principalstress space.

The test data on dense as well as loose Hostunsand are also used to study the rale of dilation.This topic was extensively treated by Bolton(1986), and it is now generally accepted that thetriaxial rate of dilation coincides with the rate ofdilation found in plane strain tests. FollowingRoscoe (t970), Bohon used an angle of dilatancyl/Jp for plane strain, but its definition is notextended to cover triaxial strain. However, anattempt at this was made by Vaid & Sasitharan(1991). A different definition is presented in thistechnical note which was previously given byVermeer & de Borst (1984) but is deriveddifferently here. Empirical evidence shows thatthe definition matches dala from both plane strainand tria"<.ial srrain.

Manuscript received II January 1995; revised manuscriptaccepted 4 May 1995.Discussion on this technical note closes 3 June 1996; forfurther details see p. ii.* Stuttgart Universi[)'.

145

LABORATORY TESTINGTriaxial compression tests were performed on a

quartz sand (Flavigny, Desrues & Palayer, 1990).This so·called Hostun sand has been used formany years (Desrues, 1984; Desrues, Colliat­Dangus & foray, 1988) in model tests and forresearch on constitutive modelling. The materialparameters were emin =0·648; emu = 1·04 I; Ps =2·65 g!cm]. Fig. 1 shows the grain size distribu­tion.

All samples were compacted by pluviation in asteel cylinder lined with a rubber membrane(to = 0·3 mm). Under a back-pressure of Uo =50 kN/m2, the samples (Ho = Do = 100 mm) wereplaced in the triaxial cell, back-pressure wasremoved, and the samples were consolidated underac• To speed up saturation, the samples weresaturated first with C02 and then with water. Thevolume change was measured by pore-watervolume change, and the specimens were axiallystrained at I% per minute.

Because the hcight-<iiameter ratio HalDo of allthe samples was unity, special means werenecessary for compensation of end restraint. Thefollowing anti-friction system was used. Both endplates (enlarged diameter 110 mm) were madefrom polished glass with a centre hole fordrainage. A silicon grease-rubber interface wasplaced between the plates and the sample. Previoustests have shown the shear parameters measuredwith this system to be equal to those measured

100

>' 80~•~~ 80~

~• 40D•E•" 20•Q.

0~063 0·125 0·25 0·5Diameter: mm

Fig. 1. Grain-size distribution of Hostun sand

146 SCHANZ AND VERMEER

conventionally with only filter plates. The presentsystem ensures a nC3r-unifonn dcfonnation of thesample up to peak stress ratio.

The bedding error 6.r c caused by the lubrica­tion, which can lead to a 60% reduction in theinitial moduli of axial stiffness. was numericallyeliminated using

,--------------,-12

5

-8

6t'fto = 0·3[1 - exp( - 0·0037oill (I)

(Goldscheider, 1982) where 10 is the thickness ofthe membrnne and 0\ is the axial stress. Also, theeffect of the lateral membrane restraint wasestimated by assuming it to be a right cylinder.With the stiffness of the membrane Em (= 1400 kNlm2

), the correction stress I!:J.03c can be calculatedaccording to

(2)

where 03 is the radial stress and E] is the radialstrain. In contrast to the bedding error, thismembrane stiffness correction had little impact onthe test results.

FRICTION ANGLESStandard drained triaxial tests were carried out

on dense Hostun s:md, with Yo = 16-3 leN/m) and10 = 1-15, and on loose Hostun sand withYo ~ 13·9 kN/mJ and I D = O·38. To check thereproducibility of test results, four control testswere performed at a fixed cell pressure ofOJ ~ 300 kN/m'.

Figures 2 and 3 shows test results; stress-straincurves are ploned with the stress ratio on the leftvertical axis and strain-strain curves are super­posed by ploning the volumetric strain on the rightvertical a,,<is. The test data show that thereproducibility of triaxial tests is quite good.

A second step in checking the reproducibilityand thus the reliability of test data is to comparedata from different laboratories. A direct compar­ison can be made between the present data (IGS)and dara from the Grenoble Institute of Mechanics(lMG) (Flavigny, Hadj-Sadok, Horodecki & Bala­chowski, 1991), as both laboratories have used thesame sand and the same testing procedure,including the lubrication of end plates. Thecomparison was made by using test data for thedense sand and plotting average values for a seriesof control tests, as shown in Fig. 4.

Even with comparable testing procedures, dif­ferent laboratories appear to produce slightlydifferent curves. Some differences with classicaltest data (aspect ratio of two and no lubrication)are expected, but the deviations between IGSresults and IMG results are surprising; as yet thereis no clear explanation. However, in terms offriction angles the differences between IGS andIMG are smaller than Fig. 4 suggests, as peak

1O~#'''--5t----:';!;Oc-----''1~5...L..'-~2;!;O,JO

(1:% I

Fig. 2. Stress-strain bchllviour of dense Hostun sand

=:5,----------_--,

-8

Fig. 3. Siress-strain behaviour of loose Hostun sand

,---------------,-12

5

-8

~3o

• IGSo IGM

.6. Non-Iubricaled

ll".~:'--'-----':---"";----:':-'Oo 5 10 15 20

[1:%

Fig. 4. Mean stress-strain bch:n'iour found in threelaboratories

friction angles of about 42° and 40° degreesrespectively are found (the precise values are givenin Table I.

Figure 4 shows that all volumetric strainscompare well up to an axial strain of about 10%,which is well beyond peak strength. Differencesoccur beyond an axial strain of 10%, when acritical state is approached in which the sample

FRICTION ANGLES AND DILATANCY OF SAND 147

(6)

(3)

(5b)-E3/EI. = D. = R./K

D=R/K

where D = -e3/e" is the stress ratio 01103 and K isa coefficient representing the internal friction whichmay be expressed as

K = lan' (45 + ¢r/2) (4)

For loose sands ¢r is equal to the friction angle¢cv at critical state, but values tend to be lower fordense sands. Rowe derived these relationships byconsidering the ratc of energy dissipation. Onchanging from plane state of strain to triaxialtesting conditions, he computcd the ratc of energyby adding the effects of two mechanisms. How~

ever, his resulting equation can also be obtainedwithout considering energy dissipation, as is nowshown. Similarly to Rowe, sliding on planesgoverned by the stress ratio 0\102 (mechanism A)and sliding on other planes governed by olloJ(mechanism B) are considered.

Figure 5 shows the A mechanism with slidingon a 0\--02 plane and the B mechanism withsliding on a 01--<73 plane. Each sliding mechanismconstitutes a planar deformation, and it is thustempting [0 apply equation (3) to each separatemechanism. This yields

-E,fEIA = DA = RA/K (Sa)

where RA = RB = Rand e2 "'" £J for triaxial testingconditions. The basic idea that follows from theseconsiderations is that there are two contributions tothe axial strain, i.e.

VALIDATION OF THE STRESS-DILATAJ<CY THEORYSeveral theories have been developed for pre­

dicting the volume strain in triaxial testing as afunction of the axial strain. In particular, theapplicabiliIy of Rowe's (1962, 1971) stress dila­tancy theory has been shown by Barden & Khayan(1966) and Wood (1990). This is also done here,but in addition Rowe's idea of superposition isemphasized as this is applied when consideringangles of dilatancy. The stress dilatancy theorystarts with the expression for plane states of strain

deforms with further change of volume. At the endof the test, at an axial strain of 17%, this criticalstate is not yet fully reached but softening anddilation are clearly damping out. At 17% verticalstrain the IGS and Th-1G data yield friction anglesof 34.80 and 35.70 respectively. It is possible that acritical state angle of almost 34-40 would havebeen reached on further straining. This angle isobtained from the loose sand data in Fig. 3, and isassumed here to be the critical state angle offriction.

Having obtained a peak friction angle of 40-420

for the dense sand and a maximum friction angleof 34-40 for the loose sand, it is interesting tocompare these triaxial angles to friction anglesmeasured in plane strain tests by Hammad (1991).The laner data are listed in Table 2 for variousvalues of the confining stress.

Taking data for a cell pressure of 300 kN/m2• aswas also done in tria"{ial testing, a peak frictionangle of 45-47° is found for the dense sand and amaximum friction angle of 32,5-34,5° for the loosesand. A significant difference is thus found for thedense sand, as other studies, whereas there is verylinle difference for the loose sand at the criticalstate. (This finding is confirmed below by data forother sands.) Hence it seems that a unique criticalstate angle ¢cv exists independently of strainconditions.

Table 1. Shear strength and dilatancy or Hostun sandunder triaxial compression

1

-""-'I "'.. I ","dcg~~es deg~~'es dewees

10 "'" 1-15

IGS 41·9 34-8 13·3

IMG 40·1 35·7 14·0

Non-lubric:ued 41·8 l7-7 12·6

10 = 0·38

IGS 34-4 34-4 I 0·0

• ,

°3: kN/m2 ",,,. degrces "'~s: degrees <pps, degrees IjIPs. degreesP . P • P •

10 • 0·95 /0 '" 0'37

100 46·7-47·5 14'5-14·7 35'5 0·0

200 46·4-47·0 14·1-14·2 32,5-34·5 0·0

400 4;·1-45·3 11·4-12·1 33·0-33·3 -1,3

Table 2. Aooles of friction and dilatanc\' of Hostun sand in the biaxial test (Ji:lmmad 1991)

148

----------SCHANZ AND VERMEER

6

Fig. 5. Deviation of triaxial dilatancy from biaxialstate

2

4

//

//

Kev (¢ev = 34'4°) ,," ",,"

//<1"='9.)/ /

/ // /

~/o L __L-__L-..!-_L-_-----1o 0'5 1 1·5 2

D =-2E:/t,

32

or in short Fig. 7. Stress-dilatancy plot for loose Hostun sand

6

(9)

lines, as the strain ratio D is computed from verysmall increments of strain. \Vhen a ratio iscomputed, small errors tend to have large con­sequences. Note that the zig-zagging would vanishif D were computed from strain increments twiceas large. In Figs 6 and 7 lines are plotted for K/..where ¢f is taken to be the interparticle angle offriction, and also for Kev , where the critical stateangle of friction is used. Accordingly to Rowe(1971), the former should be used for dense sandsand the laner is more appropriate for loose sands.However, the differences between the resultinglines is small and an average value would beadequate for most practical purposes.

ANGLE OF DILATANCYThe angle of dilatancy is first examined in plane

strain situations and its definition is then extendedto include triaxial compression. For plane strainconditions, the definition is given in severaltextbooks and by Bolton (1986)

. ps £1 +£) (8)sm¢ = --.--.-EI - E)

The first minus sign should be omitted whencontractive strains are considered positive. Whenconsidering the peak dilatancy angle rates ratherthan mobilized pre-peak angles of dilatancy, oneshould obviously use rates of strain as measured atand beyond peak stress ratios. Analogously to theextension of the stress-dilatancy theory, the conceptof a dilatancy angle can be extended to includetriaxial test conditions. Again the axial strain isconsidered to consist of an A mechanism incombination with Ez and a B mechanism thatrelates to the other principal strain E]

(7a)

(7b)

,Kev (¢ev =34'4°) "

// /

" ,,"J( (Q =29°)"" ')1 II

//2

4

D = -21:3/1:,

D=R/K

//

/ // /

// /

<­•o,L----,d-__+ __-.-',-__--!o 0'5 1 1·5 2

D= -'id~

Fig. 6. Stress-diIatancy plot for dense Hostun sand(me:m values)

Hence the difference between the plane strain(equation (3)) and equation (7a) concerns a factorof two in the definition of D, as noted by Rowe(1962). In the present derivation, the idea ofsuperposition is shown in Fig. 5, i.e. two localizedsliding motions in shear bands. In reality muchmore diffuse pre-peak deformation patterns occur,but this does not change the idea of superposing anA~type mechanism and a B-type mechanism, whichleads to the above results.

The value of the angle ,pr in the expression forK has not yet been defined. Triaxial test data arenow considered for this purpose. The data fordense and loose Hostun sand are planed in Figs 6and 7 respectively. Using equation (7) in the formR = KD, R is plotted on the vertical axis and D isplotted on the horizontal axis.

Nearly straight lines that pass through theorigin, as suggested by the expression R = KD,are found. In fact the plot zig-zags around such

FRJCTION ANGLES AND DILATANCY OF SAND 149

(10)

(II)

for triaxial srrain~ where /R is a relative dilatancyindex

BOLTON'S FINDINGS FOR PEAK ANGLESBolton (1986) assumes a unique critical state

angle tPcv for both triaxial strain and plane strain.This is confirmed by test data for Hostun sand.Bohon gives a large database which leads to thecorrelations for plane strain

</>~' - </>~ '" 51. (14)

( 15)

and

tP; - tP~v ::::: 3/R

This equation is the same as equation (12) exceptfor the superscripts, which mean that these angleshave to be measured in triaxial tests instead ofplane strain tests. In triaxial tests one' tends to findsmaller peak friction angles than in plane straintests, and Rowe reports a similar tendency for cPr.Indeed, for dense Hostun sand it is found thattPr = 29°, which is significantly different from the34·5° found earlier for ¢~s.

[t is concluded that Rowe's stress dilatancytheory exhibits an appealing relationship bet\vcenthe friction angle and the dilatancy angle forplanar deformation, in that 4>~s = ¢CII. However,this theory needs to be supplemented for triaxialconditions of stress and strain in order to obtain arelationship bet\veen the friction angle and thedilatancy angle. For this reason, relationships givenby Bolton (1986) are now considered.

Instead of combining the plane strain equations(3) and (4) of the stress-<!ilalaDcy lheory wilh thedefinition of the dilatancy angle in equation (II),one might use Rowe's equation (equation (7» fortria."{ial tests with equation (11) to obtain

. sin tPtr- sin tP7Slllrp= (13)

I - sin ¢u sin tPr

D_ D _ I - sin rp

A- B-- .I + SIn rp

This yields for t/J the expression

tv/E.Isin", =

2 - iv/E.I

Hence, a definition has been derived for thedilatancy angle that can be used to measure thisangle in triaxial compression testing. A moreformal deriyation based on concepts of the theoryof plasticity is given by Vermeer & de Borst (1984).Applying equation (II) to the triaxial test data inFigs 2 and 3, a (peak) dilatancy angle of 14° isfound for the dense Hostun sand and a vanishinglysmall value of about zero is obtained for the loosesand. These values correspond extremely well tovalues measured in plane strain tests: Hammad(1991) rep0rls virtually identical values to thosegiven in Table 2.

The plane strain definition (equation (3» for thedilatancy angle is formally equal to the triaxialdefinition (equation (11)). This is due to thefact that E2 vanishes for plane strain, givingEll = El + E3, and so equation (3) reduces to equa­tion (11). Hence the latter equation is valid forboth test conditions. This supports the finding thatIhe same dilatancy angle is measured in planestrain and triaxial tests. Bolton (1986) presentsnumerous data to show Ihal bOlh tests yield thesame peak ratio of ElllE l .

ROWE'S THEORY AND THE ANGLE OF DILATANCYThe relationship between the dilalancy angle

and the friction angle is also given by Bolton(1986). On combining the strcss-dilalancy equa­tions (3) and (4) with the definition of thedilatancy angle in equation (11), it is found that

. sin tPps- sin tPr

Sill rp = (12)I - sin tPps sin tPr

which relates density and the applied stress level. Itwas found that Q= 10 and R = 1 give the best fitfor different sands. Combining equations (14) and(15) gives

</>; '" ¥3</>:' + 2</>,") (17)

Equation (17) is not mentioned directly by Bolton,but is a direct consequence of his findings. Fig. 8provides data from additional sources.

There is a good deal of evidence for the validityof equation (17). It therefore appears that differ­ences bet\veen friction angles disappear as looser

The superscripts ps have been added to denoteplane strain angles of friction, as this formula wasderived using the plane strain equations (3) and (4),and plane strain angles of friction tend to be largerthan friction angles measured in triaxial teSIS. I asuperscript is used to denote the dilatancy angle, asthis angle is considered to be independent of testingconditions. According to Rowe cP~s coincides withthe critical state angle cPCII. If the data in Table 2 areused to compute cP~5 from equation (12), the densesand yields tP~s = 36° and the loose sand yieldstP~s = 34·5°. As the difference is relatively small,there exists .. more or less uniquely defined angle¢fPs which corresponds well with the critical stateangle.

10 = 10{Q -InUrn) - R (16)

SCHANZ AND VERMEER150

50 o Cornforth (1964)

A leussink e/af. (1966)<> Hostun sand (dense)

o Hoslun sand (loose)

,o ..,,,

,,,,,,," Equation (17)

tions. The extended theory is validated by the faelthat data from plane strain and triaxial strainconditions yield the same angle of dilatancy atleast near and beyond peak.

In contrast to the angle of dilatancy, frictionangles differ considerably when tria...ial strain andplane strains are compared. This difference basi­cally depends on the critical state friction angle, asby Bolton (1986) and other researchers. As yet itis not fully clear whelher or not plane strainconditions yield slightly higher critical state anglesthan triaxial strain conditions. Considering datafrom Hostun sand, no such difference is observed.There is linear relationship between angles ofma.'(lmum friction for both conditions (equation(17».

30 "---__.,.. -,':- ___:'~ ~ 40 ~

tisp: degrees

Fig. 8. Maximum strength under pl:me strain andtriaxial strain

states are considered. This has implications for theform of the limiting envelope for slates of stress inprincipal stress space. For dense sJrnples the planestrain friction is well above the Mohr-Coulombprediction, but looser samples give envelopesaccording to Mohr-Coulomb. There arc a lot oftrue triaxial data to confirm the former, but fewtrue triaxial tests have been performed on loosesand. Therefore it is often suggested that frictionangles are strain-dependent for both loose anddense sands. Considering results from Bolton andthe additional data of Fig. 8, the present authors donot agree.

Another finding by Bolton is that the rate ofdilation is srrain-independent. It is found for bothtriaxial strain and biaxial strain that

(18)

This suppons the idea of a unique angle ofdilatancy, as this angle was related to the aboverate of dilation. Combining equations (II) and (18)gives

ACKNOWLEDGEMENTSThe authors are indebted to Dr 1. Desrues and

Dr E. Flavigny of the Instirut de Mecanique deGrenoble for discussion on the triaxial testingtcchnique, and for their biaxial testing data onHosrun sand.

NOTATIOND diametere void ratio

Em membrane thicknessH height10 dilatancy indexIR rel:ltive dil:ltancy indexK internal friction coefficientR stress ratio (ol/oJ)to membrane thickness

!::J.r bedding errorEJ radial strainp density

00 b:lck-pressurea, axial stressOJ radial stress

1'". critical state friction angle1'p peak friction angleV'p angle of dilatancy

CONCLUSIONSFrom the results prcsented, the following con­

clusions can be drawn concerning the angles offriction and dilatancy of sand.

By using concepts of superposition it is possibleto relate the angle of dilatancy to triaxial strainconditions. This yields an extended definition forthe angle of dilatancy which applies to triaxialtesting conditions as well as plane strain condi-

. 0·3[.smllJ=

2 + 0·3[.[.

6·7 +[.(19)

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FRICTION ANGLES AND DILATANCY OF SAND 151

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